This books presents a detailed treatment of the physics of the interaction between quantum and classical variables. Semiquantum Hamiltonians represent mixed physical systems for which one part is quantal and the other classical. These parts are coupled in nonlinear fashion. The coupling term contains both quantum and classical degrees of freedom. Physical systems that are appropriately described in this way are are to be found in molecular and solid state physics. Of course, so described is the interaction between a classical measurement's instrument and the quantum system being measured. Semiquantum systems have a technological counterpart as well, in connection with nano-technology (molecular transistors, quantum dots and SQUIDS, for instance). These special systems can be successfully modelled by quantum billiards, dynamic system in which a particle alternates between motion in a straight line and elastic reflections from a boundary. The interest focuses in vibrating quantum billiards, in which a particle of mass m is subjected to elastic interactions confined in a potential well V(q), generated by a classical mass M.